# If the probability of the union of two independent equally likely events is $1$, what is the probability of each?

$E_1$ and $E_2$ are events in a probability space satisfying the following constraints:

• $\operatorname{Pr}(E_1)=\operatorname{Pr}(E_2)$
• $\operatorname{Pr}(E_1\cup E_2)=1$
• $E_1$ and $E_2$ are independent.

The probability of $E_1$ is ...(a) $0$, (b) $1/4$, (c) $1/2$, (d) $1$

I think the answer should be (d) 1

Reasoning.

1. E1 and E2 are equally likely.
2. Sum of their probability is 1. This is possible if and only if both of their probabilities are either 1(edit: if the events are independent) or 0.5( edit: if these are dependent and exhaustive events)
3. E1, E2 are independent events. This implies that they both doesn't belong to same same space.

Hence E1 and E2 are certain events with probability 1.

• See math notation guide. I replaced the image with text and formulas.
– user147263
Jan 10, 2015 at 5:36
• @Fundamental: Wow!!! Thanks. I will try to use these symbols from next time. :) Jan 10, 2015 at 11:03

Your reasoning for (2) is in error. Imagine a 3-sided die. The probability of $A=\{1,2\}$ occurring and $B=\{2,3\}$ occurring is equal and their union is a certain event.

You can get the answer quickly by elimination:

$$1=P(E_1 \cup E_2) < P(E_1)+P(E_2) = 2P(E_1) = 2P(E_2)$$

The strict less-than comes from being independent. So $P(E_1),P(E_2)$ are both larger than $1/2$

You can explicitly prove this, too

$$1=P(E_1 \cup E_2) = P(E_1)+P(E_2)-P(E_1)P(E_2) = 2P(E_1) - P(E_1)^2$$

$$\iff P(E_1)^2 - 2P(E_1)+1=(P(E_1)-1)^2=0$$

• in the example of a 3 sided die, the occurrences of any 2 events is either dependent or mutually exclusive. As per the 3rd point in the image, the events must be independent. So, I don't think we can consider this example for this question. Jan 10, 2015 at 5:01
• But your reasoning for $(2)$ without having considered their independence was in error. You didnt state that until already concluding it must be $1/2$ or $1$. You would have been marked off for this in a solution. Jan 10, 2015 at 5:03
• Thanks. I missed it. I just updated the point 2. Jan 10, 2015 at 5:13
• : according to you which option is the correct answer? Jan 10, 2015 at 5:17
• If $(x-1)^2=0$ then $x$ must be.... Jan 10, 2015 at 5:18